<li>Water is evaporating from a conical cup at the rate of 0.5 cm^3/h. The height of the cup is 10 cm, and the diameter of the top is 6 cm. How fast is the water level dropping when the water is 5 cm deep?</li>
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<p>Heres what I did:
Know: dV/dt = -0.5 cm^3/h
Want: dh/dt at h = 5
Link: V = 1/3 pi(r^2)h</p>
<p>V = 1/3 pi(r^2)h
dV/dt = 1/3 pi(2)rh dh/dt
-0.5 = 2/3 pi(1.5)(5) dh/dt
dh/dt = -0.5/(5pi) = -1/(10pi) cm^3/h</p>
<p>The correct answer is -2/(9pi) cm/h</p>
<p>Use the product rule</p>
<p>You can actually completely avoid the product rule in cases like this, since the radius and height of a cone maintain a common ratio throughout. In this case, since the radius is 3 (as the diameter is 6) when the height is 10, you know that r = (3/10h).</p>
<p>This means that V = 1/3 pi ((3h/10)^2)h, and now you don't even need the product rule.</p>
<p>One other note about the units as I look back on this:</p>
<p>Height, length, and distance is always measured in terms of linear units, while surface area and area are in square units, and volume is in cubic units.</p>
<p>Since dh/dt represents the rate of change of a linear unit, the units should be cm/h, not cm^3/h.</p>